1 / 27

Preliminaries on normed vector space

Preliminaries on normed vector space. E:normed vector space. :topological dual of E i.e. is the set of all continuous linear functionals on E. Continuous linear functional. :normed vector space. is a Banach space. Propositions about normed vector space.

jbuckman
Download Presentation

Preliminaries on normed vector space

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Preliminaries on normed vector space E:normed vector space :topological dual of E i.e. is the set of all continuous linear functionals on E

  2. Continuous linear functional :normed vector space

  3. is a Banach space

  4. Propositions about normed vector space 1. If E is a normed vector space, then is a Banach space

  5. Propositions about normed vector space 2. If E is a finite dimensiional normed vector space, then E is or with Euclidean norm topologically depending on whether E is real or complex.

  6. I.2 Geometric form of Hahn-Banach Theorem separation of convex set

  7. Hyperplane E:real vector space is called a Hyperplane of equation[f=α] If α=0, then H is a Hypersubspace

  8. Proposition 1.5 E: real normed vector space The Hyperplane [f=α] is closed if and only if

  9. Separated in broad sense E:real vector space A,B: subsets of E A and B are separated by the Hyperplane[f=α] in broad sense if

  10. Separated in restrict sense E:real vector space A,B: subsets of E A and B are separated by the Hyperplane[f=α] in restrict sense if

  11. Theorem 1.6(Hahn-Banach; the first geometric form) E:real normed vector space Let be two disjoint nonnempty convex sets. Suppose A is open, then there is a closed Hyperplane separating A and B in broad sense.

  12. Theorem 1.7(Hahn-Banach; the second geometric form) E:real normed vector space Let be two disjoint nonnempty closed convex sets. Suppose that B is compact, then there is a closed Hyperplane separating A and B in restric sense.

  13. Corollary 1.8 E:real normed vector space Let F be a subspace of E with ,then

  14. Exercise A vector subspace F of E is dence if and only if

More Related